Format and topics for final exam

Math 129a

**General information.** The final exam will be a comprehensive
timed exam, held **Fri May 17**, **12:15pm-2:30pm**, in our usual
classroom. The test will cover all of the material on the three
in-class midterms, as well as sections 5.3, 5.5, and 6.1-6.2 of the
text. No books, notes, calculators, etc., are allowed. Most of the
exam will rely on understanding the problem sets (including problems
to be done but not to be turned in), the quizzes, and the definitions
and theorems that lie behind them. If you can do all of the homework
and the quizzes, and you know and understand all of the definitions
and the statements of all of the theorems we've studied, you should be
in good shape.

You should not spend time memorizing proofs of theorems from the book,
but you should defintely spend time memorizing the *statements*
of the important theorems in the text, especially any named theorems.

**Types of questions.** The final exam will include the usual
computations, statements of definitions and theorems, true/false with
justification, and paragraph-style questions.

**Definitions.** The most important definitions and symbols we
have covered are:

5.3 diagonalizable D,Pnotation5.5 Markov chain states of a Markov chain transition matrix regular Markov chain differential equation general solution initial condition particular solution 6.1 norm (length) of v||v||distance between anduvdot product u·vorthogonal (perpendicular) orthogonal projection of onto a linev6.2 orthogonal set orthogonal basis unit vector orthonormal basis orthogonal complement W^{perp}orthogonal projection of ontovWdistance from tovW

**Examples.** Make sure you understand the following examples.

**Sect. 5.3:**- Determining if a matrix is diagonalizable (Exs. 3-4, pp. 277-279).
**Sect. 5.5:**- Finding general and particular solutions to a system of differential equations (pp. 296-297).

**Theorems, results, algorithms.** The most important theorems,
results, and algorithms we have covered are listed below. You should
understand all of these results, and you should be able to cite them
as needed. You should also be prepared to recite named theorems.

**Sect. 5.3:**- Diagonalizability Theorem (Thm. 5.2). Test for a
Diagonalizable Matrix (p. 277). A matrix with
*n*different eigenvalues is diagonalizable (p. 277). Distinct Eigenvalues Theorem (Thm. 5.3). **Sect. 5.5:**- Regular Markov chains Theorem (Thm. 5.4).
**Sect. 6.1:**- Algebraic properties of dot product and length (Thm. 6.1); Pythagorean Theorem (Thm. 6.2).
**Sect. 6.2:**- Nonzero orthogonal sets are linearly independent
(Thm. 6.5). Gram-Schmidt works (Thm. 6.6). Orthogonal complement
is a subspace (p. 327).
*(Row A)*;^{perp}=Null A*dimW+dimW*. Orthogonal projection onto^{perp}=n*W*is closest vector in*W*(p. 331). **Other:**- The Long Theorems (handout): Fat Matrix, Tall Matrix, Long Theorem.

**Types of computations.** You should also know how to do the
following general types of computations. (Note also that on the
actual exam, there will be problems that are not of these types.
Nevertheless, it will be helpful to know how to do all these types.)

**Sect. 5.3:**- Computing
*A*using^{m}*D*and*P*(pp. 270-271). Test for a Diagonalizable Matrix (p. 277) (bases of eigenspaces). Algorithm for Matrix Diagonalization (p. 277) (finding*D*and*P*). **Sect. 5.5:**- Solution to
*y'=Ay*when*A*is diagonalizable. **Sect. 6.1:**- Computing dot products, lengths, distances; are two vectors orthogonal?; orthogonal projection of a vector onto a line.
**Sect. 6.2:**- Coefficients of a vector in an orthogonal basis
(p. 322) and an orthonormal basis (p. 326). Using Gram-Schmidt to
get an orthogonal basis. Going from orthogonal to orthonormal basis.
Basis for
*W*(pp. 328-329). Orthogonal projection of^{perp}onto**v***W*(pp. 329-331).

**Not on exam.** Section 5.5: Harmonic motion, difference
equations. Section 6.1: Cauchy-Schwartz inequality, triangle
inequality.

**Good luck.**

hsu@mathcs.sjsu.edu